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Homotopy of Operads and Grothendieck–Teichmüller Groups: Part 2: The Applications of (Rational) Homotopy Theory Methods

Benoit Fresse Université de Lille 1, Villeneuve d’Ascq, France
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Hardcover ISBN: 978-1-4704-3482-3
Product Code: SURV/217.2
List Price: $135.00 MAA Member Price:$121.50
AMS Member Price: $108.00 Electronic ISBN: 978-1-4704-3757-2 Product Code: SURV/217.2.E List Price:$135.00
MAA Member Price: $121.50 AMS Member Price:$108.00
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AMS Member Price: $162.00 Click above image for expanded view Homotopy of Operads and Grothendieck–Teichmüller Groups: Part 2: The Applications of (Rational) Homotopy Theory Methods Benoit Fresse Université de Lille 1, Villeneuve d’Ascq, France Available Formats:  Hardcover ISBN: 978-1-4704-3482-3 Product Code: SURV/217.2  List Price:$135.00 MAA Member Price: $121.50 AMS Member Price:$108.00
 Electronic ISBN: 978-1-4704-3757-2 Product Code: SURV/217.2.E
 List Price: $135.00 MAA Member Price:$121.50 AMS Member Price: $108.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$202.50 MAA Member Price: $182.25 AMS Member Price:$162.00
• Book Details

Mathematical Surveys and Monographs
Volume: 2172017; 704 pp
MSC: Primary 55; Secondary 18; 57; 20;

The ultimate goal of this book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2-disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads.

The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2-disc operad.

This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory.

Graduate students and researchers interested in algebraic topology and algebraic geometry.

This item is also available as part of a set:

• Homotopy theory and its applications to operads
• General methods of homotopy theory
• Model categories and homotopy theory
• Mapping spaces and simplicial model categories
• Simplicial structures and mapping spaces in general model categories
• Cofibrantly generated model categories
• Modules, algebras, and the rational homotopy of spaces
• Differential graded modules, simplicial modules, and cosimplicial modules
• Differential graded algebras, simplicial algebras, and cosimplicial algebras
• Models for the rational homotopy of spaces
• The (rational) homotopy of operads
• The model category of operads in simplicial sets
• The homotopy theory of (Hopf) cooperads
• Models for the rational homotopy of (non-unitary) operads
• The homotopy theory of (Hopf) $\Lambda$-cooperads
• Models for the rational homotopy of unitary operads
• Applications of the rational homotopy to $E_n$-operads
• Complete Lie algebras and rational models of classifying spaces
• Formality and rational models of $E_n$-operads
• The computation of homotopy automorphism spaces of operads
• Introduction to the results of the computations for the $E_n$-operads
• The applications of homotopy spectral sequences
• Homotopy spsectral sequences and mapping spaces of operads
• Applications of the cotriple cohomology of operads
• Applications of the Koszul duality of operads
• The case of $E_n$-operads
• The applications of the Koszul duality for $E_n$-operads
• The interpretation of the result of the spectral sequence in the case of $E_2$-operads
• Conclusion: A survey of further research on operadic mapping spaces and their applications
• Graph complexes and $E_n$-operads
• From $E_n$-operads to embedding spaces
• Appendices

• Reviews

• This book provides a very useful reference for known and new results about operads and rational homotopy theory and thus provides a valuable resource for researchers and graduate students interested in (some of) the many topics that it covers. As it is the case for the first volume, careful introductions on the various levels of the text help to make this material accessible and to put it in context.

Steffen Sagave, Zentralblatt MATH
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2172017; 704 pp
MSC: Primary 55; Secondary 18; 57; 20;

The ultimate goal of this book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2-disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads.

The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2-disc operad.

This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory.

Graduate students and researchers interested in algebraic topology and algebraic geometry.

This item is also available as part of a set:
• Homotopy theory and its applications to operads
• General methods of homotopy theory
• Model categories and homotopy theory
• Mapping spaces and simplicial model categories
• Simplicial structures and mapping spaces in general model categories
• Cofibrantly generated model categories
• Modules, algebras, and the rational homotopy of spaces
• Differential graded modules, simplicial modules, and cosimplicial modules
• Differential graded algebras, simplicial algebras, and cosimplicial algebras
• Models for the rational homotopy of spaces
• The (rational) homotopy of operads
• The model category of operads in simplicial sets
• The homotopy theory of (Hopf) cooperads
• Models for the rational homotopy of (non-unitary) operads
• The homotopy theory of (Hopf) $\Lambda$-cooperads
• Models for the rational homotopy of unitary operads
• Applications of the rational homotopy to $E_n$-operads
• Complete Lie algebras and rational models of classifying spaces
• Formality and rational models of $E_n$-operads
• The computation of homotopy automorphism spaces of operads
• Introduction to the results of the computations for the $E_n$-operads
• The applications of homotopy spectral sequences
• Homotopy spsectral sequences and mapping spaces of operads
• Applications of the cotriple cohomology of operads
• Applications of the Koszul duality of operads
• The case of $E_n$-operads
• The applications of the Koszul duality for $E_n$-operads
• The interpretation of the result of the spectral sequence in the case of $E_2$-operads
• Conclusion: A survey of further research on operadic mapping spaces and their applications
• Graph complexes and $E_n$-operads
• From $E_n$-operads to embedding spaces
• Appendices